Our groups thinks we may have been doing this wrong. I thought we could only train 1 die up in a week. That is add 1 black die or upgrade a silver or gold die. One of the guys was reading the rules though and he thinks we should be able to train 2 upgrades in dice in a single week. So we could add a black die and upgrade a silver to gold in a single week (paying for each in exp and gold). Is that correct?

The rules are quite clear. You may indeed train 2 dice upgrades during one training session.